09 December 2008

On translation of games

At Language Log recent discussion has gone on about how you can translate from one language to another, but you can't translate from one game to another. For example, you can't take a game of chess and translate it into poker.

I'm reminded of the Subjunc-TV in Douglas Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid, which has characters tuning into a baseball game that has been made to look like a football game. Of course this doesn't work perfectly, which is intended to illustrate Hofstadter's points about the imperfection of analogies.

At Language Log, I learned that there are certain "logical games" for which a notion of translation is possible. These are apparently of interest to logicians; you can read more at the Stanford Encyclopedia of Philosophy.

But in combinatorial game theory, we can associate each position in certain games with a "number"; is it meaningful to say that positions in different games which have the same number are the "same position"? In this case, translations between games would become possible, except that those numbers are apparently difficult to calculate.

7 comments:

Well yes, that's kind of the point of combinatorial game theory! At least when we play games with the normal play convention (last player wins) we can substitute positions of equal value for one another without affecting the outcome of the game.

Unfortunately, from a practical point of view with respect to "games people actually play" that observation is not of too much value.

Well, Chess and Poker are ENTIRELY different sorts of games. One is a game with complete knowledge, the other isn't. Now...Chess can be translated into Nim, in theory, but as you mentioned, in practice it's awful. Poker, on the other hand, has incomplete knowledge of the position of the game for any player, and so is harder to "solve".

Well, of course, you can't translate _games_, but rather "generalized games", or more precisely, families of games parametrized by some parameter(s). For example, in Nim, they can be the number of piles, and the number of items in each one. But _the_ Nim itself (1-3-5-7 one) is not very translatable into almost anything else. Chess is just harder to parametrize, but it can be done.

Michael Albert's parenthetical note is pretty important: if you take two games with the same number under the normal play convention, and then suddenly decide to play the misère form (last player loses), the games can become very, very different from each other. So even if you consider these games equivalent, you can't necessarily change "equivalent rules" in equivalent games and expect them to stay the same.

There's also the sense of translation used in NP-completeness theory (that is, many-one polynomial time reduction). In this sense, if both generalized chess and generalized go are PSPACE-complete (with an appropriate rule for preventing more-than-polynomially-long games) then any chess position can be translated automatically into a go position with the same win-loss status and vice versa. But these translations don't respect the moves of the game, necessarily — if white moves a pawn one step, the translated go game may have nothing to do with the translation prior to the pawn move.

I'll admit I'm not a combinatorial game theorist, but I've only seen symmetric games translated into nimbers. Chess is asymmetric, since Wanda can only move the white pieces and not the black ones. The moves available to me are different than those available to Bob. Is there a way of translating asymmetric games?

Yeah. "Symmetric" and "asymmetric" games are usually referred to as "impartial" and "partisan" games, respectively, and there is quite an extensive theory about the classification of partisan games. Honestly, combinatorial game theory would be a little boring if all we cared about were the impartial ones.